Skip to main content
Log in

On the Rate of Convergence of the 2-D Stochastic Leray-\(\alpha \) Model to the 2-D Stochastic Navier–Stokes Equations with Multiplicative Noise

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

In the present paper we study the convergence of the solution of the two dimensional (2-D) stochastic Leray-\(\alpha \) model to the solution of the 2-D stochastic Navier–Stokes equations. We are mainly interested in the rate, as \(\alpha \rightarrow 0\), of the following error function

$$\begin{aligned} \varepsilon _\alpha (t)=\sup _{s\in [0,t]} |\mathbf {u}^\alpha (s)-\mathbf {u}(s)|+\left( \int _0^t |\mathrm {A}^\frac{1}{2}[\mathbf {u}^\alpha (s)-\mathbf {u}(s)] |^2 ds \right) ^\frac{1}{2}, \end{aligned}$$

where \(\mathbf {u}^\alpha \) and \(\mathbf {u}\) are the solution of stochastic Leray-\(\alpha \) model and the stochastic Navier–Stokes equations, respectively. We show that when properly localized the error function \(\varepsilon _\alpha \) converges in mean square as \(\alpha \rightarrow 0\) and the convergence is of order \(O(\alpha )\). We also prove that \(\varepsilon _\alpha \) converges in probability to zero with order at most \(O(\alpha )\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Barbato, D., Bessaih, H., Ferrario, B.: On a stochastic Leray-\(\alpha \) model of Euler equations. Stoch. Process. Appl. 124(1), 199–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bensoussan, A.: Stochastic Navier–Stokes equations. Acta Appl. Math. 38, 267–304 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bensoussan, A., Temam, R.: Equations Stochastiques du Type Navier–Stokes. J. Funct. Anal. 13, 195–222 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brzeźniak, Z., Bessaih, H., Millet, M.: Splitting up method for the 2D stochastic Navier–Stokes equations. SPDEs Anal. Comput. 2(4), 433–470 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Brzeźniak, Z., Capiński, M., Flandoli, F.: Stochastic Navier–Stokes equations with multiplicative noise. Stoch. Anal. Appl. 10(5), 523–532 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brzeźniak, Z., Peszat, S.: Strong local and global solutions for stochastic Navier–Stokes equations. Infinite Dimensional Stochastic Analysis (Amsterdam, 1999). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., vol. 52, pp. 85–98. Royal Netherlands Academy of Arts and Sciences, Amsterdam (2000)

    Google Scholar 

  7. Caraballo, T., Márquez-Durán, A.M., Real, J.: Asymptotic behaviour of the three-dimensional \(\alpha \)-Navier–Stokes model with delays. J. Math. Anal. Appl. 340(1), 410–423 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caraballo, T., Real, J., Taniguchi, T.: On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier–Stokes equations. Proc. R. Soc. Lond. Ser. A 462(2066), 459–479 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao, Y., Lunasin, E.M., Titi, E.S.: Global well-posedness of three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4, 823–848 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, Y., Titi, E.S.: On the rate of convergence of the two-dimensional \(\alpha \)-models of turbulence to the Navier–Stokes equations. Numer. Funct. Anal. Optim. 30(11–12), 1231–1271 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81, 5338–5341 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: A connection between the Camassa-Holm equations and turbulent flows in channels and pipes. Phys. Fluid. 11, 2343–2353 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, S., Foias, C., Holm, D.D., Olson, E., Titi, E.S., Wynne, S.: The Camassa-Holm equations and turbulence. Phys. D 133, 49–65 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, S., Holm, D.D., Margolin, L.G., Zhang, R.: Direct numerical simulations of the Navier–Stokes alpha model. Phys. D 133, 66–83 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, L., Guenther, R.B., Thomann, E.A., Waymire, E.C.: A rate of convergence for the LANS\(\alpha \) regularization of Navier–Stokes. J. Math. Anal. Appl. 348, 637–649 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cheskidov, A., Holm, D.D., Olson, E., Titi, E.S.: On a Leray-\(\alpha \) model of turbulence. Roy. Soc. A 461, 629–649 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chueshov, I., Kuksin, S.: Stochastic 3D Navier–Stokes equations in a thin domain and its \(\alpha \)-approximation. Phys. D 237(10–12), 1352–1367 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chueshov, I., Millet, A.: Stochastic two-dimensional hydrodynamical systems: Wong-Zakai approximation and support theorem. Stoch. Anal. Appl. 29(4), 570–611 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chueshov, I., Millet, A.: Stochastic 2D hydrodynamical type systems: well posedness and large deviations. Appl. Math. Optim. 61(3), 379–420 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press, Chicago (1988)

    MATH  Google Scholar 

  21. Deugoue, G., Sango, M.: Weak solutions to stochastic 3D Navier–Stokes-\(\alpha \) model of turbulence: \(\alpha \)-asymptotic behavior. J. Math. Anal. Appl. 384(1), 49–62 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Deugoue, G., Sango, M.: On the strong solution for the 3D Stochastic Leray-\(\alpha \) model. Bound. Value Probl. (2010). doi:10.1155/2010/723018

  23. Deugoue, G., M. Sango, M.: On the stochastic 3D Navier–Stokes-\(\alpha \) model of fluids turbulence. Abstr. Appl. Anal. (2009)

  24. Deugoué, G., Razafimandimby, P.A., Sango, M.: On the 3-D stochastic magnetohydrodynamic-\(\alpha \) model. Stoch. Process. Appl. 122(5), 2211–2248 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Flandoli, F., Gatarek, G.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Probab. Theory Relat. Fields 102, 367–391 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Foias, C., Holm, D.D., Titi, E.S.: The Navier–Stokes-\(\alpha \) model of fluid turbulence. Phys. D 152, 505–519 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Foias, C., Holm, D.D., Titi, E.S.: The three dimensional viscous Camassa-Holm equations, and their relation to the Navier–Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14, 1–35 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Geurts, B., Holm, D.D.: Leray and LANS-\(\alpha \) modeling of turbulent mixing. J. Turbul. 7, 1–33 (2006)

    Article  MathSciNet  Google Scholar 

  29. Holm, D.D., Titi, E.S.: Computational models of turbulence: the LANS-\(\alpha \) model and the role of global analysis. SIAM News 38(7) (2005)

  30. Ilyin, A., Lunasin, E.M., Titi, E.S.: A modified-Leray-\(\alpha \) subgrid scale model of turbulence. Nonlinearity 19, 879–897 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Layton, W., Lewandowski, R.: A high accuracy Leray-deconvolution model of turbulence and its limiting behavior. Anal. Appl. 6, 23–49 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Layton, W., Lewandowski, R.: On a well-posed turbulence model. Discret. Contin. Dyn. Syst. B 6, 111–128 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Leray, J.: Essai sur le mouvement d’un fluide viqueux remplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  34. Lunasin, E.M., Kurien, S., Titi, E.S.: Spectral scaling of the Leray-\(\alpha \) model for two-dimensional turbulence. J. Phys. A 41, 344014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lunasin, E.M., Kurien, S., Taylor, M., Titi, E.S.: A study of the Navier–Stokes-\(\alpha \) model for two-dimensional turbulence. J. Turbul. 8, 1–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mikulevicius, R., Rozovskii, B.L.: Stochastic Navier–Stokes equations and turbulent flows. SIAM J. Math. Anal. 35(5), 1250–1310 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mohseni, K., Kosović, B., Schkoller, S., Marsden, J.E.: Numerical simulations of the Lagrangian averaged Navier–Stokes equations for homogeneous isotropic turbulence. Phys. Fluid. 15, 524–544 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nadiga, B., Skoller, S.: Enhancement of the inverse-cascade of energy in the two-dimensional Lagrangian-averaged Navier–Stokes equations. Phys. Fluid. 13, 1528–1531 (2001)

    Article  MATH  Google Scholar 

  39. Printems, J.: On the discretization in time of parabolic stochastic partial differential equations. Math. Model. Numer. Anal. 35, 1055–1078 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  40. Temam, R.: Navier–Stokes equations and nonlinear functional analysis, 2nd edn. In: CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1995)

  41. Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

Download references

Acknowledgments

Paul Razafimandimby’s research was funded by the FWF-Austrian Science Fund through the Project M1487. Hakima Bessaih was supported in part by the Simons Foundation Grant #283308 and the NSF Grants DMS-1416689 and DMS-1418838. The research on this paper was initiated during the visit of Paul Razafimandimby at University of Wyoming in November 2013 and was finished while Paul Razafimandimby and Hakima Bessaih were visiting KAUST. They are both very grateful to both institutions for the warm and kind hospitality and great scientific atmosphere.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hakima Bessaih.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bessaih, H., Razafimandimby, P.A. On the Rate of Convergence of the 2-D Stochastic Leray-\(\alpha \) Model to the 2-D Stochastic Navier–Stokes Equations with Multiplicative Noise. Appl Math Optim 74, 1–25 (2016). https://doi.org/10.1007/s00245-015-9303-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-015-9303-7

Keywords

Mathematics Subject Classification

Navigation